Question

Geometric Mean question

Answer 1

\[\frac{ 8\sqrt{2} }{ 3 }\times \frac{ 4\sqrt{2} }{ 3 }\]

Answer 2

@SolomonZelman @campbell_st @tkhunny

Answer 3

every time I do it I get \[24\sqrt{2}=12\sqrt{2}\]

Answer 4

How did you spawn an equal sign? What is it you are trying to do?

Answer 5

I have no idea what I'm doing honestly

Answer 6

its geometric mean so\[\frac{ a }{ x } = \frac{ x }{ b }\]

Answer 7

its geometric mean so\[\frac{ a }{ x } = \frac{ x }{ b }\]\[x = \sqrt{ab}\]

Answer 8

Suppose, I want to find the "geometric mean" between \(4\) and \(12\).
(I am going to use \(r\) to denote the "geometric mean")

I am assuming you are familiar with "geometric sequences"
So, basically, this \(r\) must be a term between \(4\) and \(12\), in
a pattern of a geometric sequence.

+++++

In other words: \(\color{#000000 }{ \displaystyle 4,~~r,~~12 }\)
must be a geometric sequence.

Answer 9

okay.

Answer 10

Actually, let's use m for geometric mean ....

Answer 11

i know the answer is 8/3 I just need to know how to get there

Answer 12

So,
You can choose to conceptualize it this way:

You are given that a sequence is geometric,
and you are given two terms, which are as follows:
\(\color{#000000 }{ \displaystyle a_1=4 }\)
\(\color{#000000 }{ \displaystyle a_3=12 }\)

and you are asked to find the second term,
\(\color{#000000 }{ \displaystyle a_2=m }\)
(We know that second term is \(m\), but we need
to find this \(m\).)

Just applying the basics of geometric sequence, we have:
\(\color{#000000 }{ \displaystyle a_3=a_1\times r^{3-1} }\)
\(\color{#000000 }{ \displaystyle 12=4\times r^{2} }\)
\(\color{#000000 }{ \displaystyle 3=r^{2} }\)
\(\color{#000000 }{ \displaystyle \sqrt{3}=\sqrt{r^{2}} }\)
\(\color{#000000 }{ \displaystyle \sqrt{3}=|r|}\)
\(\color{#000000 }{ \displaystyle r=\pm\sqrt{3}}\)

So \(\color{#000000 }{ \displaystyle r=\pm\sqrt{3}}\) is the common ratio between \(\color{#000000 }{ \displaystyle a_1}\) and \(\color{#000000 }{ a_2}\).

\(\color{#000000 }{ \displaystyle a_2=a_1\times r }\)
\(\color{#000000 }{ \displaystyle m=4\times (\pm \sqrt{3}) =\pm 4\sqrt{3} }\)

Answer 13

I think thats way more complicated than it needs to be.

Answer 14

I am just giving an example, and writing out EVERYTHING, but I can say this much more shortly, if you like.

Answer 15

Short version:
++++++++

\(\color{#000000 }{ \displaystyle m }\) - geometric mean
\(\color{#000000 }{ \displaystyle s }\) - some number


There is an \(\color{#000000 }{ \displaystyle m }\), such that:

(1) \(\color{#000000 }{ \displaystyle m\times s=x }\)
and, (2) \(\color{#000000 }{ \displaystyle m\div s=z }\)

Answer 16

For example:

The geometric mean between \(\color{#000000 }{ \displaystyle 2 }\) and \(\color{#000000 }{ \displaystyle 18 }\) is \(\color{#0000ff }{ \displaystyle 6 }\).

Because,
\(\color{#000000 }{ \displaystyle 18\div 3=6 }\)
\(\color{#000000 }{ \displaystyle 2\times 3=6 }\)

That is that \(\color{#0000ff }{ \displaystyle 6 }\) is a "term of a geometric sequence" between 2 and 18.

Answer 17

how does this fit into geometric mean?

Answer 18

what do you mean?

Answer 19

how do I answer my problem?

Answer 20

I've never heard of geometric sequence, the question is about geometric mean.

Answer 21

oh, sorry, ... the two concepts really go together, so I thought you have learned about geom. seq. before.

Answer 22

That is, that Y is "geometrically" or "multiplication-wise" is right between W and Z.


hopefully it starts to make more sense.....